After trying several things to get it to work as required, (in particular a change recommended on another blog site to change the 403 error to redirect to https://exchange.domain.com/owa), I ended up breaking access to the Exchange Management Console and Exchange Management Shell in Exchange 2010 with the following error:
Connecting to remote server failed with the following error message: The WinRM client cannot process the request. It cannot determine the content type of the HTTP response from the destination computer. The content type is absent or invalid.
The issue was with IIS and therefore caused users using Outlook to receive prompts to re-authenticate to Exchange. Also Out of Office replies, Free/Busy information, mailtips and the ability to move emails into subfolders all stopped working.
The eventual fix was to restore IIS settings. Copy, paste and replace the Applicationhost.CONFIG file from either backup media or the IIS auto backups located in c:\inetpub\history into c:\windows\system32\inetsrv\config (Recommend you back up this file first before replacing).
You will also need to restore (copy, paste and replace) from backup media the web.CONFIG file in c:\inetpub\wwwroot
Restart IIS services and you should now be able to get into Exchange Management Console and Shell.